Lectures on stochastic sewing with applications

Abstract: These are lecture notes for a mini-course on stochastic sewing, taught at the University of Edinburgh and Beijing Institute of Technology in Spring/Summer 2025. The aim is to introduce the reader to stochastic sewing techniques and to show how they can be successfully applied to study various problems in stochastic analysis, including: regularization by noise for stochastic differential equations driven by Brownian motion or fractional Brownian motion, well-posedness of stochastic PDEs with irregular drift, the study of averaging operators and local times, and the analysis of numerical algorithms.